Due Mar 23, 3:59 AM EDT
How many vertices are there in the biggest clique in this graph?

Indeed, the pentagon (5 vertices on the left) is a clique, while the whole graph (6 vertices) is not.
How many vertices are there in the biggest independent set in this graph?

Vertices c, d, e, f form an independent set. An independent set of 5 vertices is impossible, since it would include at least two vertices in the a-b-c triangle.
How many vertices could a maximal clique in a bipartite graph include?
A bigger clique would include a triangle, which is impossible in a bipartite graph.
A bipartite graph has 6 vertices and 9 edges. How many vertices does the biggest independent set in this graph include?
Such a graph must have 3 vertices in one part and 3 vertices in the other one. (If there are 4 and 2, the total number of edges is less or equal than 8, if 5 and 1, it is less or equal than 5. Both are less than 9.) Moreover, any pair of vertices from different parts is connected. Thus, these parts are exactly maximal independent sets.